Question: Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find $z^{2000}+\frac 1{z^{2000}}$.
Solution: From the given equation, $z^2 + 1 = 2z \cos 3^\circ,$ or $z^2 - 2z \cos 3^\circ + 1 = 0.$  Then by the quadratic formula,
\[z = \frac{2 \cos 3^\circ \pm \sqrt{4 \cos^2 3^\circ - 4}}{2} = \cos 3^\circ \pm i \sin 3^\circ.\]Then by DeMoivre's Theorem,
\[z^{2000} = \cos 6000^\circ \pm i \sin 6000^\circ = \cos 240^\circ \pm i \sin 240^\circ,\]and
\[\frac{1}{z^{2000}} = \cos (-240^\circ) \pm i \sin (-240^\circ) = \cos 240^\circ \mp i \sin 240^\circ,\]so
\[z^{2000} + \frac{1}{z^{2000}} = 2 \cos 240^\circ = \boxed{-1}.\]